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This study focuses on presenting a new approach to studying the robust mean-square exponential stability of uncertain Markovian jump systems with mode-dependent time-varying delays. The basic idea of this approach is to choose distinct Lyapunov matrices for different system modes. To achieve it, a novel Lyapunov functional is constructed with the novelty being that: (i) besides Pi,Q1i,Q2i,Q3i, the Lyapunov matrices R1i,R2i of double-integral terms depend on the system mode i and (ii) two additional double-integral terms are introduced to resolve the difficulties brought by the terms with R1i,R2i. Some less conservative conditions are derived such that the Markovian jump system is robustly mean-square exponentially stable for all admissible uncertainties. It is further rigorously shown that some recent results are the special cases of the stability criterion established by the new approach. An illustrative example is given to show the performance of the developed results.